infinitesimal

Both the infinitely large and its inverse, the infinitely small, cause headaches. The first philosophical explorations that turned disquiet into something more tangible were Zeno's paradoxes . The 18th-century development of the calculus led to the consideration of magnitudes changing over ‘indefinitely’ smaller intervals of time, and the reaction on Zeno's behalf came from Berkeley, who saw that the infinitesimals resulting (‘the vanished ghosts of departed quantities’) were treated inconsistently as finite quantities or as nothing, depending on the exigencies of the argument. A rigorous treatment of the notion of a limit was only provided by K. T. W. Weierstrass (1815–97) in the 19th century: it need only refer to finite numbers, and thus removes the place of the infinitesimal from the foundations of mathematics. However, a consistent non-standard analysis has been developed by the mathematician Abraham Robinson (1918–74): in it infinitesimals have the disturbing property that while they are not nothing, no sum of n of them is ever greater than any finite number, so the number sequence that includes them is not Archimedean.